The voltage across an air-filled parallel-plate capacitor is...

The voltage across an air-filled parallel-plate capacitor is measured to be 85.0 V. When a dielectric is inserted and completely fills the space between the plates as in Figure P26.42, the voltage drops to 25.0 V. (a) What is the dielectric constant of the inserted material? (b) Can you identify the dielectric? If so, what is it? (c) If the dielectric does not completely fill the space between the plates, what could you conclude about the voltage across the plates?

Key Concepts

Constant Charge versus Constant Voltage Conditions

In capacitor problems, it is crucial to identify whether the charge on the capacitor remains constant or if the voltage is held constant when modifications (such as inserting a dielectric) are made. In a constant-charge scenario, any increase in capacitance due to the dielectric leads to a decrease in voltage across the plates, whereas in a constant-voltage regime, the amount of stored charge would change instead.

Dielectric Constant (Relative Permittivity)

The dielectric constant, or relative permittivity, is a dimensionless quantity that compares the capacitance of a capacitor using a given dielectric material to the capacitance of the same capacitor with a vacuum between its plates. It is a measure of how effectively the dielectric material can increase the capacitance by reducing the electric field within the capacitor.

Partial Dielectric Insertion Effects

When a dielectric does not completely fill the space between the plates, the system behaves like a composite capacitor with regions having different dielectric properties. This arrangement results in an effective capacitance that is intermediate between that of a capacitor fully filled with the dielectric and one completely filled with air. Consequently, the voltage across the plates and the overall capacitance are affected by the proportion of the gap that is occupied by the dielectric.

Parallel-Plate Capacitor

A parallel?plate capacitor consists of two conductive plates separated by a small distance, and its ability to store electrical energy (capacitance) depends on the area of the plates, the distance between them, and the properties of the material in between. It serves as a basic element in circuits, providing a model for understanding how charge, voltage, and stored energy interact in the presence or absence of additional materials.

Dielectric Material

A dielectric is an insulating material placed between the plates of a capacitor that increases the capacitance by reducing the effective electric field. When a dielectric is inserted, the material becomes polarized in the presence of the electric field, which partially cancels the field generated by the charges on the plates, allowing the capacitor to store more charge at a given voltage.

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